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The knee of a curve can be defined as a vertex of the graph. This corresponds with the graphical intuition (it is where the curvature has a maximum), but depends on the choice of scale. The term "knee" as applied to curves dates at least to the 1910s, [1] and is found more commonly by the 1940s, [2] being common enough to draw criticism.
A saddle point (in red) on the graph of z = x 2 − y 2 (hyperbolic paraboloid). In mathematics, a saddle point or minimax point [1] is a point on the surface of the graph of a function where the slopes (derivatives) in orthogonal directions are all zero (a critical point), but which is not a local extremum of the function. [2]
A turning point of a differentiable function is a point at which the derivative has an isolated zero and changes sign at the point. [2] A turning point may be either a relative maximum or a relative minimum (also known as local minimum and maximum). A turning point is thus a stationary point, but not all stationary points are turning points.
A critical point of such a curve, for the projection parallel to the y-axis (the map (x, y) → x), is a point of the curve where (,) = This means that the tangent of the curve is parallel to the y -axis, and that, at this point, g does not define an implicit function from x to y (see implicit function theorem ).
At this point, called the maximum sustainable yield, there is a surplus of individuals that can be harvested because growth of the population is at its maximum point due to the large number of reproducing individuals. Above this point, density dependent factors increasingly limit breeding until the population reaches carrying capacity.
A diagram illustrating great-circle distance (drawn in red) between two points on a sphere, P and Q. Two antipodal points, u and v are also shown.. The great-circle distance, orthodromic distance, or spherical distance is the distance between two points on a sphere, measured along the great-circle arc between them.
For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature at a point of a differentiable curve is the curvature of its osculating circle — that is, the circle that best approximates the curve near this ...
Low-order polynomials tend to be smooth and high order polynomial curves tend to be "lumpy". To define this more precisely, the maximum number of inflection points possible in a polynomial curve is n-2, where n is the order of the polynomial equation. An inflection point is a location on the curve where it switches from a positive radius to ...